Advertisements
Advertisements
Question
If α and β are the zeros of the quadratic polynomial f(x) = ax2 + bx + c, then evaluate α4 + β4
Advertisements
Solution
f(x) = ax2 + bx + c
α + β = `(-b/a)`
αβ = `c/a`
since α + β are the roots (or) zeroes of the given polynomials
then
α4 + β4 = (α2 + β2)2 -2α2 + β2
= ((α + β)2 - 2αβ)2 - 2(αβ)2
`=[(-b/a)^2-2c/a]^2-[2(c/a)^2]`
`=[(b^2-2ac)/a^2]^2-(2c^2)/a^2`
`=((b^2"2ac")^2-2a^2c^2)/a^4`
APPEARS IN
RELATED QUESTIONS
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients
`f(x)=x^2-(sqrt3+1)x+sqrt3`
If α and β are the zeros of the quadratic polynomial f(x) = ax2 + bx + c, then evaluate α2β + αβ2
If α and β are the zeros of the quadratic polynomial f(x) = ax2 + bx + c, then evaluate :
`a(α^2/β+β^2/α)+b(α/β+β/α)`
If If α and β are the zeros of the quadratic polynomial f(x) = x2 – 2x + 3, find a polynomial whose roots are α + 2, β + 2.
Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and product of its zeros as 3, −1 and −3 respectively.
If the zeros of the polynomial f(x) = x3 − 12x2 + 39x + k are in A.P., find the value of k.
Find the zeroes of the quadratic polynomial f(x) = 4x2 - 4x - 3 and verify the relation between its zeroes and coefficients.
Find the zeroes of the polynomial f(x) = `2sqrt3x^2-5x+sqrt3` and verify the relation between its zeroes and coefficients.
Find the zeroes of the quadratic polynomial `(5y^2 + 10y)` and verify the relation between the zeroes and the coefficients.
Find the quadratic polynomial, sum of whose zeroes is 8 and their product is 12. Hence, find the zeroes of the polynomial.
If f(x) = `x^4– 5x + 6" is divided by g(x) "= 2 – x2`
On dividing `3x^3 + x^2 + 2x + 5` is divided by a polynomial g(x), the quotient and remainder are (3x – 5) and (9x + 10) respectively. Find g(x).
If 𝛼, 𝛽 are the zeroes of the polynomial f(x) = x2 + x – 2, then `(∝/β-∝/β)`
Define a polynomial with real coefficients.
If α, β, γ are are the zeros of the polynomial f(x) = x3 − px2 + qx − r, the\[\frac{1}{\alpha\beta} + \frac{1}{\beta\gamma} + \frac{1}{\gamma\alpha} =\]
What should be subtracted to the polynomial x2 − 16x + 30, so that 15 is the zero of the resulting polynomial?
The polynomial which when divided by −x2 + x − 1 gives a quotient x − 2 and remainder 3, is
Basketball and soccer are played with a spherical ball. Even though an athlete dribbles the ball in both sports, a basketball player uses his hands and a soccer player uses his feet. Usually, soccer is played outdoors on a large field and basketball is played indoor on a court made out of wood. The projectile (path traced) of soccer ball and basketball are in the form of parabola representing quadratic polynomial.
What will be the expression of the polynomial?
If α, β are zeroes of quadratic polynomial 5x2 + 5x + 1, find the value of α–1 + β–1.
Find the zeroes of the quadratic polynomial 4s2 – 4s + 1 and verify the relationship between the zeroes and the coefficients.
